Physics Lesson 16.6.4 - Magnetic Field Inside a Long Straight Wire with Current

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Welcome to our Physics lesson on Magnetic Field Inside a Long Straight Wire with Current, this is the fourth lesson of our suite of physics lessons covering the topic of Ampere's Law, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Magnetic Field Inside a Long Straight Wire with Current

In this case, the radius r of Amperian loop is smaller than the radius R of wire. This means the Amperian loop is taken inside the wire's section. Physics Tutorials: This image provides visual information for the physics tutorial Ampere's Law

Like in the previous paragraph, the distribution of magnetic field has a cylindrical symmetry as the current is uniformly distributed throughout the wire (the direction is out of page). Therefore, we use a similar approach as in the calculation of magnetic field outside the wire discussed in the previous paragraph.

Using Ampere's law for this situation, we obtain for r < R:

∮B⃗ dL⃗ = ∮B ⋅ cos⁡θ dL
= B ∙ ∮dL
= B ∙ (2π ∙ r)
= μ₀ ∙ i_encl

Since the current distribution is uniform, the current enclosed in the loop is proportional to the area of loop at any point, i.e.

i_encl/i_tot = π ∙ r²/π ∙ R²

i_encl = i_tot ∙ π ∙ r²/π ∙ R²

Substituting this value for current in the previous formula obtained after integrating Ampere's law equation, yields

B ∙ (2π ∙ r) = μ₀ ∙ i_tot ∙ π ∙ r²/π ∙ R²

B = μ₀ ∙ i_tot/2π ∙ R² ∙ r

The above formula means the magnetic field is zero at centre of cross section of the wire and it has a maximum value for r = R (on the outer surface of wire). This is another proof that the electric charges (that are responsible for the magnetic field generation) are concentrated on the outer layer of conductor, as we have explained in Section 14.

You have reached the end of Physics lesson 16.6.4 Magnetic Field Inside a Long Straight Wire with Current. There are 5 lessons in this physics tutorial covering Ampere's Law, you can access all the lessons from this tutorial below.

More Ampere's Law Lessons and Learning Resources

Magnetism Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
16.6	Ampere's Law
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
16.6.1	Useful Mathematical Background: Integral of a function - Geometrical approach
16.6.2	Ampere's Law
16.6.3	Magnetic Field Outside a Long Straight Wire with Current
16.6.4	Magnetic Field Inside a Long Straight Wire with Current
16.6.5	Ampere's Law Applied in Solenoids and Toroids

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